Pierre Lelong

Entire Functions of Several Complex Variables

1. Measures of Growth.- 1. Preliminaries.- 2. Subharmonic and Plurisubharmonic Functions.- 3. Norms on ?n and Order of Growth.- 4. Minimal Growth: Liouvilles Theorem and Generalizations.- 5. Entire Functions of Finite Order.- 6. Proximate Orders.- 7. Regularizations.- 8. Indicator of Growth Functions.- 9. Exceptional Sets for Growth Conditions.- Historical Notes.- 2. Local Metric Properties of Zero Sets and Positive Closed Currents.- 1. Positive Currents.- 2. Exterior Product.- 3. Positive Closed Currents.- 4. Positive Closed Currents of Degree 1.- 5. Analytic Varieties and Currents of Integration.- Historical Notes.- 3. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set.- 1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor.- 2. Indicators of Growth of Cousin Data in ?n.- 3. Canonical Potentials in ?m.- 4. The Canonical Representation of Entire Functions of Finite Order.- 5. Solution of the ?

A Class of Fréchet Complex Spaces in which the Bounded Sets are C-Polar Sets

bar partial

Holomorphic Mappings from ℂ n to ℂ m

Equation.- 6. The Case of a Cousin Data.- 7. Slowly Increasing Cousin Data: the Genus q = 0 the Algebraic Case.- 8. The Case of Integral Order: Extension of a Theorem of Lindelof.- 9. Trace of a Cousin Data on Complex Lines.- 10. The Case of a Cousin Data of Infinite Order.- Historical Notes.- 4. Functions of Regular Growth.- 1. General Properties of Functions of Regular growth.- 2. Distribution of the Zeros of Functions of Regular Growth.- Historical Notes.- 5. Holomorphic Mappings from ?n to ?m.- 1. Representation of an Analytic Variety Y in ?n as F-1(0).- 2. Local Potentials and the Defect of Plurisubharmonicity.- 3. Global Potentials.- 4. Construction of a System F of Entire Functions such that Y=F-1(0).- 5. The Case of Slow Growth.- 6. The Algebraic Case.- 7. The Pseudo Algebraic Case.- 8. Counterexamples to Uniform Upper Bounds.- 9. An Upper Bound for the Area of F-1(a) for a Holomorphic Map.- 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes.- Historical Notes.- 6. Application of Entire Functions in Number Theory.- 1. Preliminaries from Number Theory.- 2. A Schwarz Lemma.- 3. Statement and Proof of the Main Theorem.- Historical Notes.- 7. The Indicator of Growth Theorem.- Historical Notes.- 8. Analytic Functionals.- 1. Convex Sets and the Fourier-Borel Transform.- 2. The Projective Indicator.- 3. The Projective Laplace Transform.- 4. The Case of M a Complex Submanifold of ?n.- 5. The Generalized Laplace Transform and Indicator Function.- 6. Support for Analytic Functionals.- 7. Unique Supports for Domains in ?n.- 8. Unique Convex Supports.- Historical Notes.- 9. Convolution Operators on Linear Spaces of Entire Functions.- 1. Linear Topological Spaces of Entire Functions.- 2. Theorems of Division.- 3. Applications of Convolution Operators in the Spaces Ep?(r)and Eo.- 4. Supplementary Results for Proximate Orders with ?>1.- 5. The Case ?=1.- 6. More on Functions of Order Less than One.- 7. Convolution Operators in ?n.- Historical Notes.- Appendix I. Subharmonic and Plurisubharmonic Functions.- Appendix II. The Existence of Proximate Orders.

Functions of Regular Growth

in which a and b are complex numbers, transforms the formal development F= Ei,j aijxiyi into a power series Fa,b(t) with a nonvanishing radius of convergence, the series F= Ei I aijxiyil converges for sufficiently small I xj and I yI. The conclusion is, in other terms: F is the Taylor series of a function of two complex variables x, y which is analytic at the point (x=O, y=O). M. A. Zorn suggested, and Rimhak Ree [2] proved recently, that it is sufficient to consider, in the hypothesis, the substitutions Sab with a and b real. This result is however incomplete: we are here resolving the following problem: what are the minimal conditions which must be imposed on a class [Sab] of substitutions (1.1) transforming F into a convergent power series Fa,b(t), in order that the preceding conclusion hold for any given formal development F? It is convenient, for our purpose, to introduce the following definition. DEFINITION. The class [Sa,b] of substitutions (1.1) is called normal if the convergence of the power series Fa,b(t) for every SabE [Sa,b] implies for any given formal development F to be the Taylor series of an analytic function of x, y. The following remarks will be useful: if X is a complex number different from zero, and a ==Xa, j3 =Xb, then the series Fao(t) and Fab(t) are simultaneously convergent or divergent; the substitution S00 is degenerate and we may suppose that it is excluded from [Sa,b]. We consider therefore the correspondences

The Indicator of Growth Theorem

Publisher Summary This chapter presents a class of Frechet complex spaces in which the bounded sets are C-polar sets. Given a complex topological vector space E (written shortly c.t.v.s.), the complex analysis on E has to use two different kinds of classes of sets that interfere in E: (1) the classes of sets that are defined by the topology only, such as compact, pre-compact, bounded sets and (2) the classes of sets whose definition needs more and is related to the analytic structure on E: such classes are invariant by the complex analytic isomorphisms. Examples are analytic sets, C-polar sets, and negligible sets. These classes are defined with reference to holomorphic or plurisubharmonic functions. For complex analysis, it is often necessary to compare the two kinds (1) and (2) of notions.

The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set

We will let ℂ represent the field of complex numbers and ℝ the subfield of real numbers. Let z=(z1,…,z n ) be an element of ℂ n and ℝ 2n , the underlying space of real coordinates. The transformations from the complex to the real coordinates are given by ({z_k} = {x_k} - i{y_k},,,,{bar z_k} = {x_k} - i{y_k}) and ({x_k} = frac{{{Z_k} + {{bar Z}_k}}}{2},{y_k} = frac{{{Z_k} - {{bar Z}_k}}}{{2i}}).

Convolution Operators on Linear Spaces of Entire Functions

We shall study four problems here related to entire mappings defined on ℂ n . If X is a Cousin data in ℂ n , we have seen in Chapter 3 that we can define X as the zero set of an entire function f whose growth is related to the growth of v x (r), the indicator of X. Our first task will be to show a similar property for analytic varieties Y of arbitrary co-dimension in ℂ n . We shall show that we can define Y as Y = {z: F(z) = 0} where F: ℂ n →ℂ n +1 and the growth of ǁFǁ is bounded by v y (r), the projective indicator of growth of the current of integration on the analytic set Y (cf. Definition 2.24).

Application of Entire Functions in Number Theory

We have seen in Chapter 3 that there is a relationship between the asymptotic growth of the quantity M f (r) for an entire function f and the area of the zero set of f. In certain cases, however, much more can be said. We shall prove here the fundamental principle for functions of finite order and of regular growth, which, paraphrased a little crudely, states that an entire function of finite order has its zero set “regularly distributed” asymptotically if and only if it has “regular growth” asymptotically along all rays. An equivalent formulation, as we shall see, is to say that for an entire function f, r−ρ(r) log ∣f (r z)∣ converges (as a distribution) in L loc 1 (ℂ n ) to h f ⋆ (z) if and only if r−ρ(r)Δlog ∣ f(r z)∣ converges as a distribution to Δh f ⋆ (z). These ideas of regularity will be made more precise below.